ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Viewing a small DFA as an NFA with singleton transitions

Example

Consider the two-state DFA over ${0, 1}$ with states $e, o$ , start state $e$ , accepting state $e$ , and transitions that toggle between $e$ and $o$ on input $1$ and stay fixed on input $0$ .

Regarded as an NFA, the same machine has singleton one-letter transition sets and no epsilon-transitions. It still recognizes the language of words with an even number of $1$ 's.

Facts & Assumptions

Given: The parity DFA just described.

[L1]

By Every DFA is an NFA, every DFA can be viewed as an NFA by replacing each one-letter transition by a singleton transition set and adding no $ε$ -moves.

Verification

technique · direct

1.1

The parity DFA has exactly one next state on each input symbol, so the associated NFA has transition sets ${e}$ or ${o}$ and never branches.

given

2.1

Because there is only one live branch on every input, the NFA reaches exactly the same state as the DFA after every word. Therefore it accepts exactly the words with an even number of $1$ 's.

L1step 1.1

3.1

This is a concrete instance of the DFA-to-NFA embedding from [L1].

L1step 2.1∎

Depends on

Every DFA is an NFA

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jean Gallier and Jocelyn Quaintance, Introduction to the Theory of Computation: Some Notes for CIS511 (standard reference, not scraped)
John Watrous, Introduction to the Theory of Computing, Lecture 3: Nondeterministic finite automata (standard reference, not scraped)